Such observables always evolve slowly comparing to the torque transport, independently of the contour-length (Text S5). A collective observable \(\mathbf { X}\)  is associated with a certain amount of free energy, with its gradient along \(\mathbf { X}\)  as a driving force to reach the free energy minima[1]}. Based on this idea we derive below the free energy and the dynamic equations associated with the slow variable \(\mathbf { X}\) .
